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SZ|DG|[Jxj6i&dtf̠ˁ阭OL Ry6NcI1J*SL0l/.{IMxa,[1kYSf\z: 9_SQL{ f -4#Ehzj}Fz@aU{1Kgn+uxIqkf {~&m֝B!kC\*K/fz92nȈH圪gAڏ6׮;Y Ø?5fMl(㽦dKRʓ;:4#b;E@1t& v'=m\љ&S!//~̻CIqiz)[}纖Wmܥt:;Bw.}nLIIJtKC 8b\, ײ庿l@]^AӺVS~=x ,V[(6ZoGQn/( +J;F se| j4"_>5|{2G﷫,fr~U.aNs2~^ Y 6g=?H>ԍU!R*xhaGwLݑ 32dU@upa0/EsA(ZMԃm5q؉!zL]PAQ7wHt*M^~n"g +ܘobiЅa"^ˇ6RO c_?iX")gx4q9E&]5B}T)^iÅ*H[,M8i]^ZϛOv{ژ݃}*zv-*tq"VԍS T*JfmlWMo$u*MYy[9N뤇.QVY=&OӪd^՝ +=)a-^f/<*!+rGU;QƓ5,}T7GB-guF`{eDԋe*GBY^+2}?ch].NwW}(:8l _Js ++/QO~D\2ϴ!Hlr7k$~JXYQPka؄SIrUу hK>QUD[4LA٥:Om1ũTN?JOoi?Vn87%xZ;?|c`V WQAӛ(.#Ycn[2/5caL*Oݫ.d^CtOd9ξм8QEr}1[4GNɮZ[!sVk>;aJWl:zy_΃vCDgo,vs쵆Ʀ3[67Ln)]K +ZZl#z($ + &?_*3e +|xo:ڨO:P >Yl5G #a ZJc0!oYE5_gқmM>%9JHd8>hEaN='Y7z=.P0NQHx]=,dՠAVezt*+ տo$8wc^.y, V)04,5xieuW W +6\3?ĥR|y%tͣd0r_s we.@|%w9'caI#2Ï ^J\~O/ds*)kd|9}V>f+98vZ,|W^ +^XTBeE32>$k⒓VI3SrjK|]Z`u CX:{-F^wab*g'?'+ܿWYE]gVqZBfS-Ӯ~M OX^l +>u^h{r)̠ӺkxcsXt>Oh,FVG$fۮRԝs l5er쐪Cڼ w_z6D^]`Vt0{1~ɝ28ѧb#6œGNJ9R[T̗P]3fD<w!p4yx!A^Ł.`X豣34uEZo +vʾj9)Tq=9?klE5~YӸchΙ +j1 6 w^ʽ/: 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riassuntiva/main.tex @@ -2058,9 +2058,48 @@ rango, allora $V$ e $W$ sono isometrici tra loro (come conseguenza stavolta del teorema di Sylvester complesso). - \subsection{Operatori simmetrici, ortogonali, hermitiani e unitari} + \subsection{Trasposta e aggiunta di un'applicazione} + + Sia $(V, \varphi)$ uno spazio dotato di un prodotto $\varphi$ non degenere. Allora si definisce $f^* \in \End(V)$ (talvolta indicato come $f^\top$ se $\varphi$ non è hermitiano, quando è chiaro che non ci stia riferendo alla trasposizione dell'operatore $f$) + come l'unico operatore tale per cui $\varphi(f^*(\v), \w) = \varphi(\v, f(\w))$. + In particolare, se $\varphi$ non è hermitiano, tale operatore soddisfa la seguente relazione: + \[ \alpha_\varphi \circ f^* = f^\top \circ \alpha_\varphi, \] + + dove con $f^\top$ si indica l'applicazione trasposta di $f$. Scelta allora una + base $\basis$ di $V$, sempre se $\varphi$ non è hermitiano, si può scrivere in relazione a $M_\basis(f)$ la + matrice associata a $f^*$: + \[ M_\basis(f^*) = M_\basis(\varphi)\inv M_\basis(f)^\top M_\basis(\varphi). \] + + Se invece $\varphi$ è hermitiano, vale la seguente relazione: + \[ M_\basis(f^*) = M_\basis(\varphi)\inv M_\basis(f)^* M_\basis(\varphi). \] + + Se $\varphi$ è un prodotto scalare, $f^* = f^\top$ si chiama \textit{trasposto} + di $f$, mentre se $\varphi$ è hermitiano $f^*$ si dice \textit{aggiunto} di $f$. + + D'ora in poi si intenderà con $f^\top$ il trasposto di $f$ (con $\varphi$ scalare) e con $f^*$ l'aggiunto di $f$ (con $\varphi$ hermitiano). - [TODO] + \begin{itemize} + \item $(f^\top)^\top = f$, + \item $(f^*)^* = f$, + \item $(\lambda f)^\top = \lambda f^\top$, + \item $(\lambda f)^* = \conj{\lambda} f^*$, + \item $(f + g)^\top = f^\top + g^\top$, + \item $(f + g)^* = f^* + g^*$, + \item se $f$ è invertibile, $(f^\top)\inv = (f\inv)^\top$ (è sufficiente mostrare che $\varphi((f^\top \circ (f\inv)^\top)(\v), \w) = \varphi(\v, \w)$ e dedurre, + sottraendo i due membri, che deve valere $f^\top \circ (f\inv)^\top = \Idv$), + \item se $f$ è invertibile, $(f^*)\inv = (f\inv)^*$ (come sopra), + \item $\Ker f^\top = (\Im f)^\perp$, + \item $\Ker f^* = (\Im f)^\perp$, + \item $\Im f^\top = (\Ker f)^\perp$, + \item $\Im f^* = (\Ker f)^\perp$, + \item se $W$ è un sottospazio di $V$, allora $W$ è $f$-invariante se e solo + se $W^\perp$ è $f^*$-invariante, + \item l'operatore $\top \in \End(\End(V))$ è sempre diagonalizzabili + e ha spettro $\{\pm 1\}$, dal momento che il suo polinomio minimo è $x^2-1$ (infatti $(f^\top)^\top = f$ e gli autospazi relativi a $1$ e $-1$ sono entrambi diversi da $\zerovecset$), + \item l'autospazio $V_1$ di $\top$ raccoglie gli operatori simmetrici, mentre + $V_{-1}$ raccoglie gli operatori antisimmetrici. + \end{itemize} + \subsection{Spazi euclidei reali e complessi} @@ -2394,7 +2433,7 @@ \subsection{Classificazione delle coniche} Classificazione \textit{affine}: si trova una forma canonica nelle orbite del gruppo affine $A_n(\KK)$. - se $V$ è euclideo su $\RR$, si può introdurre la distanza $d(P,Q)=\norm{Q-P}$. + Se $V$ è euclideo su $\RR$, si può introdurre la distanza $d(P,Q)=\norm{Q-P}$. Le isometrie di $E$ sono quelle affinità che preservano la distanza e formano il gruppo $ISO(E)$. Classificazione \textit{isometrica}: Si trovano forme canoniche nelle orbite rispetto a $ISO(E)$ %???